1s - EE 478 Handout #8 Multiple User Information Theory...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EE 478 Handout #8 Multiple User Information Theory October 7, 2008 Homework Set #1 Solutions 1. Solution: (a) Consider H ( X | Z ) H ( X,Y | Z ) = H ( Y | Z ) + H ( X | Y,Z ) H ( Y | Z ) + H ( X | Y ) . (b) h ( X + Y ) h ( X ). Since conditioning reduces the differential entropy, h ( X + Y ) h ( X + Y | Y ) = h ( X ) . (c) I ( g ( X ); Y ) I ( X ; Y ). I ( g ( X ); Y ) = H ( Y )- H ( Y | g ( X )) H ( Y )- H ( Y | X ) = I ( X ; Y ) . (d) I ( Y ; Z | X ) I ( Y ; Z ) if p ( x,y,z ) = p ( x ) p ( y ) p ( z | x,y ). Since X and Y are independent random variables, I ( Y ; Z | X ) = H ( Y | X )- H ( Y | Z,X ) = H ( Y )- H ( Y | Z,X ) H ( Y )- H ( Y | Z ) = I ( Y ; Z ) . 2. Solution: (a) Consider h ( Y n ) n X i =1 h ( Y i ) = n X i =1 1 2 log(2 e ) K ii = 1 2 log(2 e ) n n Y i =1 K ii . 1 (b) Now since h ( Y n ) = 1 2 log(2 e ) n | K | , the result in part (a) implies that | K | n Y i =1 K ii . 3. Solution: The identity can be proved by induction, or more simply using chain rule of mutual information. Notice that by chain rule the mutual information between Z n and W can be expanded in n ! ways, depending on how we order the element of Z n . For instance, these are both valid expansions I ( Z n ; W ) = n X i =1 I ( Z i ; W | Z i- 1 ) = n X j =1 I ( Z j ; W | Z n j +1 ) , where the two ordering Z n = ( Z 1 ,...,Z n ) and Z n = ( Z n ,...,Z 1 ) have been used. Therefore, we have n X i =1 I ( X n i +1 ; Y i | Y i- 1 ) = n X i =1 n X j = i +1 I ( X j ; Y i | Y i- 1 ,X n j +1 ) (1) = n X j =2 j- 1 X i =1 I ( X j ; Y i | Y i- 1 ,X n j +1 ) (2) = n X j =2 I ( X j ; Y j- 1 | X n j +1 ) (3) = n X j =1 I ( X j ; Y j- 1 | X n j +1 ) (4) = n X i =1 I ( Y i- 1 ; X i | X n i +1 ) where (1) and (3) follow from chain rule of mutual information, (2) is obtained switch- ing the order in the summations, and finally (4) follows from the fact that Y = ....
View Full Document

Page1 / 7

1s - EE 478 Handout #8 Multiple User Information Theory...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online